Few More Spin/ Angular Momentum Questions

I’ve got a few more conceptual questions on spin etc, any help would be appreciated.

Reading books (eg. Sakuri) it seems like authors tend to show there’s a homomorphism between the groups SO(3) and SU(2) using Euler angles etc. I know the Pauli matricies act as generators for the group SU(2), so does this and the homomorphism automatically mean the Pauli matricies can be considered to be generators of infinitesimal rotations and so lead to a conservation law (spin) because of rotational invariance? Also why do books seem to do it this way around rather than showing an isomorphism between the generators of SO(3) and SU(2) (which I suspect exists)?

One more thing, if the Pauli matricies do act as generators of rotations why do we get both orbital angular momentum and spin, when it seems like there’s only really one unique set of generators, but then we have two conserved quantities: spin and orbital angular momentum?

Thanks in advance.

This post has been edited by plmokn on Oct 18 2007, 10:29 AM

I'm about to head out so I don't have time for a long reply sorry.

Yes, because they have the same algebraic struction. If memory serves, SU(2) has a double covering of SO(3), in that there's two elements in SU(2) which correspond to an element of SO(3).

It's more convenient to do it in the group format due to the nice 2-1 relation between group elements. That wouldn't be so clear if you did it via the generators but it would still be possible.

While spin and angular momentum are both generated by the same set of generators, they are not connected. It's L = J+S, they form seperate group structures.

More technically, the state of angular momentum is going to be the tensor product of the spin state and the orbital angular momentum state. The generators for one are algebriaic the same as the generators as the other, but they don't use one another. In other words, you'd have two seperate set of generators.

For spin you'd have {σ_1, σ_2, σ_3}, which obey [σ_i,σ_j] = 2i e_ijk σ_k, and for orbital angular momentum you'd have {Σ_1,Σ_2,Σ_3} which obey [Σ_i,Σ_j] = 2i e_ijk Σ_k and then the additional property that [Σ_i,σ_j]=0. They are seperate from one another. True, you can't get this to work in 2x2 matrices but if you think of them as tensor producted states, you end up withing in block diagonal 4x4 matrices.

I'll go into detail when I have more time.

On further thinkings, I thought I'd elaborate on some things :

Both SU(2) and SO(3) have the same Lie algebra structure (ie their generators obey the same equation) and thus their generators are somewhat equivalent, though obviously one set will be 2x2 complex matrices and the other 3x3 real.

If you were just working abstractly though, with symbols, and weren't careful, you might make the mistake of thinking they are the same Lie group. However, this is only true locally. It's possible to go from algebra to group via the exponentiation map. This would give you the same group structure near the identity but not globally.

It turns out that while L(SO(3)) is isomorphic to L(SU(2)), SU(2) and SO(3) are only homomorphic. Through a rather long and painful proof involving taking SO(3) and SU(2) to have the topological structure of S³, you find that SO(3) is homomorphic to SU(2)/Z_2, where the Z_2 symmetry equates M and -M in SU(2).

SU(2) is nicer to consider because it's simply connected, while SO(3), as outlined in this page, is not simply connected. It's common that an SU(n) group will be a nice replacement for an SO(m) group (of same dimension) because of the fact complex numbers allow for smooth paths through the group. For instance, in the Reals you cannot go from -1 to 1 without going through 0. In C, you just move in a curved path. Not quite valid as an analogy, but it's the rough idea. Another such example of SO(3,1) and SL(2,C). SO(3,1) is the Lorentz group. Well actually, part of SO(3,1) is the Lorentz group, it has 4 seperate sections (determined by it's det(L) = -+1 and weather L_00 > 1 or L_00 < -1), so if you're working with SO(3,1) you have to only work with part of it. However, SL(2,C) has the same Lie algebra structure but acts as a complete cover for it, there's no messing around with 4 discrete chunks. It's SL(2,C) which forms the structure of Weyl spinors, which you'll come across next year probably.

I hope that's helped a bit more. It's an area I struggled on when I first came across it but it's had 2~4 years (depending on which part) to bubble away at the back of my mind and that helps a great deal, so stick with it even if it's all a bit confusing at present

Cool, thanks.